 Euler–Maclaurin formula

The Euler-Maclaurin formula relates a finite sum to an integral and endpoint derivatives. If $a<b$ are integers, $p\ge 1$ and $f$ has continuous derivatives through order $2p$ on $[a,b]$, thenHere $B_{2r}$ are Bernoulli numbers. With ${\tilde{B}}_{2p}(x)=B_{2p}(\{x\})$ the periodic Bernoulli function, the exact remainder isThis form rearranges the last endpoint correction in the NIST statement.^[ref]

For $f(x)=x^2$ on $[0,n]$ and $p=1$, the remainder vanishes because the second Bernoulli polynomial has integral zero over each unit interval. The result is ${\sum }_{k=1}^n{k}^2=n^3/3+n^2/2+n/6$, the square pyramidal number formula. For other functions, retaining or bounding $R_p$ makes the finite truncation a controlled approximation.

The Euler–Maclaurin formula is a powerful mathematical tool that provides a connection between discrete sums and continuous integrals. It is useful in various areas of numerical analysis, calculus, and asymptotic analysis. The formula allows us to approximate sums by integrals, compensating for the differences with correction terms.